Electrodynamic models of 2D materials: can we match thin film and single sheet approaches?
Bruno Majérus, Evdokia Dremetsika, Michaël Lobet, Luc Henrard, Pascal Kockaert
aa r X i v : . [ phy s i c s . op ti c s ] J u l Electrodynamic models of 2D materials: can we match thin filmand single sheet approaches?
Bruno Majérus , Evdokia Dremetsika , Michaël Lobet , Luc Henrard , and PascalKockaert ∗21 Department of Physics & Namur Institute of Structured Matters (NISM), University ofNamur, 61 rue de Bruxelles, B-5000 Namur, Belgium. OPERA-photonics, Université libre de Bruxelles (U.L.B.), 50 Avenue F. D. Roosevelt, CP194/5, B-1050 Bruxelles, Belgium John A. Paulson School of Engineering and Applied Sciences, Harvard University, 9Oxford Street, Cambridge, MA 02138, United States of AmericaJuly 10, 2018
Abstract
The electromagnetic properties of 2D materials aremodeled either as single sheets with a surface sus-ceptibility or conductivity, or as thin films of finitethickness with an effective permittivity. Their in-trinsic anisotropy, however, has to be fully describedto reliably predict the optical response of systemsbased on 2D materials or to unambiguously inter-pret experimental data. In the present work, we com-pare the two approaches within the transfer matrixformalism and provide analytical relations betweenthem. We strongly emphasize the consequences of theanisotropy. In particular, we demonstrate the crucialrole of the choice of the thin film’s effective thick-ness compared with the parameters of the single sheetapproach and therefore the computed properties ofthe 2D material under study. Indeed, if the isotropic thin film model with very low thickness is similar toan anisotropic single sheet with no out-of-plane re-sponse, with larger thickness it matches with a singlesheet with isotropic susceptibility, in the reasonablesmall phase condition. We illustrate our conclusions ∗ Corresponding author:
[email protected] on extensively studied experimental quantities suchas transmittance, ellipsometry and optical contrast,and we discuss similarities and discrepancies reportedin the literature when using single sheet or thin filmmodels.
The electromagnetic (EM) properties of 2D materi-als are at the forefront of the present research ac-tivities. Further developments for applications as di-verse as optical modulators, transparent conductivefilms, photovoltaic systems, superabsorbers or sen-sors request an accurate description of the electro-magnetic response [1, 2, 3, 4]. For example, opticalcontrast or transmission are among the commonlyused quantities to characterize 2D systems, in par-ticular, to determine their thickness or their numberof layers [5, 6, 7, 8, 9]. Furthermore, electromag-netic properties are the macroscopic fingerprints ofelementary excitations such as the inter-band transi-tion, excitons or plasmons. Their correct analysis istherefore crucial for the understanding of the under-1ying physics of 2D materials.Several models have been recently used in this con-text. The EM response to an external field has beenfirstly considered as a purely 2D phenomenon withthe definition of a single sheet (surface) conductiv-ity σ s , or susceptibility χ s [10, 11]. In particular,for graphene, an analytical expression for σ s basedon tight-binding approximation and Kubo formulahas become popular [12] and provides a clear dis-tinction between inter-band and intra-band electronictransitions. The surface conductivity can be deter-mined experimentally, e.g. via Brewster angle mea-surements [13].Using another approach, 2D materials have beenconsidered as isotropic materials with a small but fi-nite thickness [7, 8, 9, 14, 15]. This approach notablyallows to use the well-developed transfer matrix tech-nique to predict and interpret optical data (includingellipsometry) with widely available methodology andnumerical codes. The thickness is often arbitrarilytaken as the interlayer distance of the 3D counterpartof the 2D material [11, 16], considered as a fitting pa-rameter [7] or evaluated based on the variation of theelectronic density in the transverse direction [17].However, these two approaches (a purely 2D sur-face conductivity and a 3D isotropic thin film) donot match as demonstrated analytically and numer-ically [18, 19] and give model-dependent interpreta-tion of ellipsometric data [20]. This is particularlytrue for oblique incidence and TM ( p -polarised) EMradiation [21]. Indeed, considering only a purely in-plane 2D conductivity means that the out-of-plane re-sponse of the layer is neglected, while for an isotropicthin film, both the in-plane and out-of-plane re-sponses are linked. Very recently, a criterion has beenproposed to determine in which conditions the twomodels give similar results at normal incidence [22].Anisotropic thin films have also been studied. Theout-of-plane component has been taken as a free pa-rameter [20, 23, 24], or deduced from first princi-ple approach calculations performed with periodicboundary conditions [19, 25]. The out-of-plane sus-ceptibility in a single sheet model has been recentlyconsidered by two of us to analyze the non-linear op-tical response of graphene [26]. An adequate descrip-tion of the out-of-plane component is of prime neces- single sheet thin film χ s ε f ab afb O y xz~k aF ~k aB ~k bF ~k bB F a B a F b B b d a d b d f Figure 1: Schematic representation of the two config-urations. Left: current sheet model. Right: thin filmwith an effective material f extending over a distance d a (resp. d b ) on the a (resp. b ) side. The wave vec-tors of the forward F and backward B fields in media a and b are denoted by ~k a,bF,B . The reference frame is Oxyz .sity since very diverse 2D materials with potentiallylarge out-of-plane polarisabilities are synthesized [27]or predicted [28].In this work, we study analytically and numericallythe conditions on the EM response function (surfacesusceptibility, dielectric tensor) and on the thicknessof the effective thin film for a correct description ofthe response of 2D materials. In particular, we an-alytically link the surface susceptibility of the singlesheet to the ordinary and extraordinary optical con-stants of the equivalent thin film. We then focus ourattention on the determination of the surface conduc-tivity of the 2D materials based on the interpretationof optical transmission, ellipsometry and optical con-trast measurements.
In this section, we perform the comparison betweenthe single sheet and the thin film approaches withinthe framework of transfer matrix formalism for strati-fied media [29, Sec. 4.6]. As a first step, we build thetransfer matrix of a single sheet at the interface oftwo surrounding media (respectively a and b ), as de-picted on Fig. 1 (left). In a second step, we calculatethe transfer matrix of a thin film with finite thick-ness d f Fig. 1 (right). We then analytically compare2he two approaches and highlight the consequenceson quantities that can be easily determined experi-mentally (transmittance, ellipsometric data, opticalcontrast). Importantly, we insist here on the conse-quences of the intrinsic anisotropy of 2D materials.The single sheet is described by a surface sus-ceptibility tensor χ s diagonal in our reference frame(Fig. 1). The in-plane components of the 2D materialare directly related to its surface conductivity by σ sα = − i ε ωχ sα , (1)where α = x, y . The out-of-plane component of thesusceptibility χ sz is also considered here, but an out-of-plane conductivity has no physical meaning for asingle sheet.The thin film material is described by a dielectrictensor ε f related to the bulk conductivity compo-nents by ε fαβ = (cid:18) ε + iσ α ω (cid:19) δ αβ . (2)The in-plane and bulk conductivities are related by σ sα = d f σ α [19]. The incident and substrate mate-rials ( a and b ) can be anisotropic but with their op-tical axes aligned with those of the 2D material, i.e. ε a,bαβ = ε ε a,bα δ αβ , which is the case in most (if not all)the systems studied experimentally so far. This hy-pothesis avoids the coupling between transverse elec-tric (TE) and transverse magnetic (TM) modes. Weallow those surrounding materials to have a complexpermittivity, and express the dependence in the an-gle of incidence via the wavevector ~k = ( k x , k y , k z ) . Ifthe incident medium is a perfect dielectric character-ized by the real isotropic permittivity ε ε a = ε n a ,and k y = 0 , we have k x = k n a sin α i , with k thewavenumber of the light in vacuum, n a the refractiveindex of medium a , and α i the angle of incidence. In order to describe the out-of plane component inthe single sheet model, we use the approach describedin [26], based on [30, 31]. In particular, the transmis-sion coefficient t and the reflexion coefficient r of theelectric field in TE and TM configurations can be Table 1: Definition of the coefficients in TE and TMconfigurations, to the first order in ϕ x , ϕ y , ϕ z . Inthese expressions, m and n will be replaced by a , b and f to denote respectively the incidence medium,the substrate and the thin film. The forward (resp.backward) component F m (resp. B m ) is defined ineach medium m with respect to the forward (resp.backward) component of the electric field parallel tothe interface [ E mx,y ] F (resp. [ E mx,y ] B ), with y for TE,and x for TM.TE (s-polarization) TM (p-polarization) k ω/ck x in-plane component of input ~kk mz q ε my k − k x p ε mx ( k − k x /ε mz ) F m [ E my ] F ε mx /k mz [ E mx ] F B m [ E my ] B − ε mx /k mz [ E mx ] B t F b /F a r B a /F a α mn k nz /k mz ( ε mx k nz ) / ( ε nx k mz ) t mn / (1 + α mn ) r mn t mn − ϕ x k az k bz ε bx k az + ε ax k bz χ sx ϕ y k k az + k bz χ sy ϕ z k x ε bx k az + ε ax k bz ε ax ε bx ǫ ab χ sz ǫ ab /ε az +1 /ε bz ) ϕ ± ϕ x ± ( ϕ y + ϕ z ) χ sjj i σ sjj / ( ε ω ) written as t = t ab [1 + i( ϕ x + ϕ y + ϕ z )] , (3) r = t − − ϕ x , (4)with the parameters defined in table 1. We note that t ab , which is the transmission coefficient in absenceof 2D material, depends on the propagation directionand is therefore not symmetrical, i.e. t ba = 2 − t ab ,while ϕ x , ϕ y , ϕ z do not depend on the propagationdirection.In these notations the transfer matrix between theincident medium a and the outgoing medium (sub-3trate) b can be written as S ab = 1 t ab · (cid:18) − i ϕ + r ab + i ϕ − r ab − i ϕ − ϕ + (cid:19) , (5)so that the forward ( F ) and backward ( B ) field com-ponents in media a and b at the single sheet interfaceare linked by (cid:18) F a B a (cid:19) = S ab (cid:18) F b B b (cid:19) . (6)The expressions for TE and TM modes have a similarform if the forward and backward components aredefined as in table 1. The matrix S ab includes theout-of plane response of the current sheet χ sz through ϕ z , and can therefore be compared to the thin filmmodel. We present in this section the propagation in the ef-fective thin film system of thickness d f described bythe diagonal tensor ε f of components ε fx , ε fy , and ε fz .The total transfer matrix of the thin film ( T ab ) in-volves the transfer matrix at the two interfaces ( I af and I fb ) and the propagation matrix P f in the ho-mogeneous film f over a distance d f . Then T ab = I af P f I fb (7)with P m = (cid:18) e − iΦ m
00 e iΦ m (cid:19) , (8) I mn = 1 t mn (cid:18) r mn r mn (cid:19) , (9)where Φ m = k mz d m , d m is the thickness of layer m ,and k mz , r mn and t mn are defined in table 1. Theanisotropy of the media is described through the di-agonal components of the dielectric tensors. The two models are considered equivalent if theirtransfer matrices are identical. However, we cannotdirectly compare S ab and T ab since the propagation in the slab of thickness d f = d a + d b is not consideredin S ab . The correct equality is then T ab = P a S ab P b . (10)In the limit of small phase shift, we can develop(10) to the first order in k d f for the bulk parameters( Φ a , Φ b , Φ f ≪ ), and to the first order in k χ s for thesingle sheet parameters ( ϕ x , ϕ y , ϕ z ≪ ). A lengthybut straightforward calculation provides the effectivedielectric function of the thin film as ε fx = χ sx /d f + η a ε ax + η b ε bx , (11) ε fy = χ sy /d f + η a ε ay + η b ε by , (12) ε fz = η a ε az + η b ε bz − χ sz ǫ ab d f , (13)where η a = d a /d f , and η b = d b /d f and then η a + η b =1 (Fig. 1). As expected, the effective dielectric tensorcomponents do not depend on the angle of incidenceangle. Nevertheless, those quantities depend on the2D material through χ s , and on the geometry of thethin film defined by d a and d b . More surprisingly, thecomponents of the dielectric tensor of the surround-ing materials ε a and ε b also appear. In the frequentcase where the incident medium is air, and the thinfilm of thickness d f is lying on top of the substrate b ,we have d a = d f , d b = 0 and ε ai = 1 , so that ε fx = χ sx /d f + 1 , (14) ε fy = χ sy /d f + 1 , (15) ε fz = 1 + 1 + ε bz ε bz d f χ sz . (16)Equations (14) and (15) are commonly used for 2Dmaterials and perfectly valid under the assumptionsreported above. The relation for the out-of-planecomponents, (13) and (16), are far from being in-tuitive but are important to understand the link be-tween the isotropic thin film and the anisotropic sin-gle sheet models. Indeed, they explain some discrep-ancies between the two approaches reported in theliterature, as we will discuss later. In the absence ofout-of-plane susceptibility ( χ sz = 0 ), (16) gives ε fz = 1 and the effective thin film is anisotropic.4 Discussion
In the following section we compare the results of thetwo approaches on quantities that are easily obtainedexperimentally: transmittance, ellipsometry and op-tical contrast.
In TM configuration, from (3) and table 1, the changein transmittance induced by the 2D material in thesmall phase shift hypothesis (first order in k d f ∼ k χ s ) and for real ε a and ε b is (cid:12)(cid:12)(cid:12)(cid:12) tt ab (cid:12)(cid:12)(cid:12)(cid:12) − − t ab k bz ε bz Im χ sx (cid:20) k x k az k bz ε ax ε bz ǫ ab (cid:18) Im χ sz Im χ sx (cid:19)(cid:21) (17) = − t ab k bz ε bz Im ε fx d f " k x k az k bz ( ε ax ) ε bz | ε fz | (cid:18) Im ε fz Im ε fx (cid:19) . (18)The change in transmittance (17) is then only relatedto Im χ s and, via (1), to Re σ s . Simple transmittancemeasurements can therefore not provide informationon Re χ s or Im σ s . In contrast, both the real andthe imaginary parts of ε fz are present in (18) through | ε fz | .To understand the impact of using the thin filmmodel instead of the single sheet approach, and totest the validity range of (11)–(13) with respect to k χ s , we have performed extensive numerical simula-tions. All the numerical results presented here are fora TM wave incident on an air/2D/glass ( n b = 1 . )system with an angle θ = 75 °, and a thickness d f = 0 . nm, except otherwise specified.Fig. 2 displays the difference of transmittance ( ∆ T )for two incident wavelengths (in the IR, λ = 1550 nm,and in the visible, λ = 700 nm) obtained with the sin-gle sheet model with no out-of-plane susceptibility( χ sz = 0 ) and with the anisotropic thin film with ε f from (11)-(13). The transmittance computed in the anisotropic thin film model and in the single sheetmodel are obviously in very good agreement. There-fore, in the following, we will consider that the single sheet model and the anisotropic thin film model giveequivalent results. This rationalizes also the fact thatthe small phase shift condition is satisfied for a largerange of 2D susceptibilities.When the small phase shift condition is relaxed( | Re [ k χ s ] | ' ), a small discrepancy can be ob-served, corresponding to the yellow bands on the sidesof Fig. 2(a) and (b). A comparison between these twofigures shows that larger discrepancies are observedat λ = 700 nm, than at λ = 1550 nm. For a betterinterpretation of the data, we identify on the figurepossible values for graphene conductivity based onthe Kubo formula [12]. In this case, we observe aparticularly small ∆ T with maximum of · − % .Interestingly, an isotropic thin film model showsmore important discrepancies when compared withan anisotropic thin film, as illustrated on Fig. 3 evenif, for a large range of values, both isotropic andanisotropic thin film models provide very similar re-sults. Note that the scale of ∆ T is different on Fig.2 and Fig. 3. For Fig. 3, ε x and ε y are equal andobtained from (11),(12) and ε x = ε y = ε z .Notably, a high value of ∆ T is observed on a ver-tical line corresponding to Re [ χ s ] = d f , for whichthe real part of the permittivity ε = 1 − χ s /d f van-ishes. This shows, similarly to what was reportedin [21], that an artificial plasmonic resonance is pre-dicted by an isotropic thin film model, due to the ar-tificial metallic nature of the out-of-plane componentof the permittivity tensor. This unphysical resonancecould have dramatic effects on the prediction of theoptical properties.To investigate further the influence of theanisotropy, we present in Fig. 4 the difference be-tween the transmittance obtained with the isotropicand anisotropic thin film models in TM configurationfor graphene, as a function of the incident wavelengthand of the thin film thickness. The two models givevery similar results for a ratio λ/d f > (dashedline) ( i.e. very small k d f ). This validates the factthat the anisotropy of graphene has been often dis-regarded without consequences on the validity of theconclusions.This surprisingly good predictions within theisotropic thin film model for intrinsically anisotropic2D material is explained as follows. The isotropic5 ) b) Figure 2: Relative difference ∆ T between the transmittance computed in the single sheet and in theanisotropic thin film models, with respect to the real and imaginary part of the single sheet susceptibil-ity ( k χ sx ). The system considered is air/graphene/glass. The refractive index of glass is taken as . . (a)Infrared EM radiation ( λ = 1550 nm); (b) Visible light ( λ = 700 nm). Circled areas indicate the range ofvalues k χ sx for graphene within the Kubo formula for a range of Fermi level from . eV to eV and arange of relaxation time from fs to fs. a) b) Figure 3: Same as Fig. 2 for the isotropic thin film model.6igure 4: Difference of transmittance ∆ T betweenthe isotropic and anisotropic thin film of graphene fora system air/graphene/glass. The refractive index ofglass is taken as . . Graphene is modeled using theKubo formula with E F = 0 . eV, τ = 100 fs. The reddotted line represent a thickness equal to / ofthe wavelength.thin film model does not correspond to the assump-tion χ sz = χ sx , but to ε fz = ε fx . By means of (13), thisis equivalent to set Im χ sz = ε ax ǫ ab | ε fz | d f Im ε fz = ε ax ǫ ab | χ sx /d f | Im χ sx , (19)which shows that the isotropic thin film model tendsto the anisotropic single sheet one with χ sz = 0 when | χ sx | ≫ d f , as in this case (19) provides χ sz ≈ .This justifies that the isotropic thin film model canbe used with good results to model graphene with Im[ χ sz ] = 0 , and d f = 0 . nm. In particular, it con-firms that in the limit d f = 0 , both models agree, asreported in [32]. More importantly, this also resolvesthe apparent contradiction between the conclusionsof [32], and those of [18, 19, 21] on the equivalence (ornot) of both models for d f → . Indeed, (19) showsthat, if we model graphene by means of a thickerlayer so that χ sx ≪ d f ≪ /k , the isotropic thin filmmodel corresponds to the isotropic single sheet one( χ sx = χ sz in the particular case were n a = n b = 1 )and no more to the anisotropic single sheet with χ sz = 0 . We illustrate this analytical observation onFig. 5, where we plot the bracket in (17) and (18) for PSfrag repla ements In iden e angle (°) R e du e d tr a n s m i tt a n e χ sz = 0 χ sz = χ sz ǫ fz = ǫ fx , d f = 0 . nm ǫ fz = ǫ fx , d f = 5 nm Figure 5: Reduced transmittance [brackets in (17)and (18)] for and air/graphene/glass structure: n a =1 , n b = 1 . , χ sx = (1 .
50 + 2 . at a wavelength of nm [15]. Curves are for: anisotropic single sheetwith no out-of-plane response ( χ sz = 0 ); isotropic thinfilm model ( ε fz = ε fx ) with d f = 0 . nm and d f =5 nm; isotropic current sheet model ( χ sz = χ sx ).an air/graphene/glass system at 634 nm as functionof the angle of incidence for different approaches: thesingle sheet for χ sz = 0 (full line) and with ( χ sx = χ sz )(dashed line), the isotropic thin film for d f = 0 . nm(dot-dashed line) and d f = 5 nm (crosses). The re-sults for the anisotropic thin film cannot be distin-guished from those of the anisotropic single sheet, andare therefore not plotted. The thickness d f = 5 nm iscommonly used in discrete numerical simulations toavoid prohibitive numerical cost [21]. As expected,at normal incidence, all curves are superimposed andthe anisotropy does not play any role. As the an-gle of incidence increases, the z -component of theresponse functions becomes more important and theexact value of the thickness of the thin film influencesthe computed optical properties.Although transmittance change at different angleswould in principle allow to separate the in-planeand the out-of-plane responses of the imaginary partof the susceptibility (real part of the conductivity),these measurements are usually performed at normalincidence, for which the TE and TM cases coincide.7 .2 Ellipsometry Ellipsometry records the ratio of the reflexion ortransmission of a sample in TM and TE configura-tions, at different angles. From equations (3) and(4), still in the small phase approximation ( i.e. tothe first order in k χ s ), we get ρ t ρ t = t T M t T Mab t T Eab t T E = (cid:2) (cid:0) ϕ T Mx − ϕ T Ey + ϕ T Mz (cid:1)(cid:3) , (20)with ρ t the ellipsometric ratio with the 2D material,and ρ t the same ratio for the interface without 2Dmaterial. Under the assumption that χ sx = χ sy , usingtable 1, we can write ρ t ρ t − − i k x k az + k bz (cid:18) χ sx − ε ax ε bx ǫ ab χ sz (cid:19) . (21)The in-plane χ sx and the out-of plane χ sz susceptibili-ties can then not be separated by means of standardtransmission ellipsometry as the coefficient in frontof χ sz is independent of k x and k z , and therefore ofthe angle of incidence.In order to analyze further these results, it is con-venient to define the parenthesis in (21) as χ ell = χ sx − ε ax ε bx ǫ ab χ sz , (22)that contains all the dependence in the susceptibility.Note that the same dependence is found for reflexionellipsometry.We can here provide a simple explanation to thedifference reported in [20] between the susceptibili-ties extracted from ellipsometric data using differentmodels for MoS on glass substrate. We named [ χ sx ] i the in-plane susceptibility deduced from the isotropicthin film model with d f = 0 . nm (Fig. 6, bluecurve with × ) and [ χ sx ] a the one found with theanisotropic single sheet model (Fig. 6, red curve with (cid:3) ).For the isotropic thin film, using (16), χ ell = [ χ sx ] i (cid:18) − ε bx χ sx ] i /d f (cid:19) , (23)while for the anisotropic sheet, we simply get χ ell = [ χ sx ] a . (24) PSfrag repla ements 300 400 500 600 700 800 900-505101520 R e [ χ s x ] ( n m ) Wavelength λ (nm)PSfrag repla ements300400500600700800900-505101520 Re[ χ sx ] (nm)Wavelength λ (nm) 400 800 λ (nm)-1.25-1.35-1.45 R e [ ∆ χ s x ] ( n m ) Figure 6: Values of
Re[ χ sx ] for MoS single layer re-trieved with the isotropic thin film model (x) and theanisotropic current sheet model ( (cid:3) ). Data from [20].The ( ⋄ ) curve is calculated from ( × ) using the shiftcalculated from (25) ( + in the inset); ( ∆ ) is (x)shifted by − ε bx d ( o in the inset). The dielectric func-tion of N-BK7 glass (substrate used in [20]) is takenfrom Sellmeier’s equation provided by Schott [33].Comparing the last two equations, we obtain [ χ sx ] a = [ χ sx ] i − ε bx d f + ε bx d f χ sx ] i /d f . (25)The inset of Fig. 6 displays the difference between [ χ sx ] i and [ χ sx ] a and compares it with − ε bx d f . Wesee that the last term in (25) is negligible. We donot reproduce the imaginary part of the susceptibility(real part of the conductivity), as it is not affected bythe real shift − ε bx d f , as is visible in [20].This good agreement between experimental dataand the analytical predictions again confirms the va-lidity of the small phase shift approximation ( k χ s ≪ ).We conclude that standard ellipsometry providesno information on the x − z anisotropy of the 2D sam-ple. It is however important to note that the singlesheet approach imposes implicitly χ sz = 0 , while theisotropic thin film approach assumes χ sz = ǫ ab χ sx /ε fx ,which explains differences reported in the literature,8or the retrieval of χ sx from ellipsometric data. Equa-tion (22) allows to introduce in the fitting procedurea value for χ sz based on theoretical assumptions orobtained experimentally, for example, by means ofcontrast ratio measurements. The optical contrast of 2D materials on a thick sub-strate is often very small and hardly measurable.However, reflexion microscopy and optical contrastare commonly used to determine the presence of 2Dmaterials (or the number of layers) if a thin dielectricfilm is lying on top of the substrate, most often
SiO on Si [5, 6, 34]. This additional layer creates interfer-ences that depend on the 2D susceptibility and allowto tune the total reflectance of the system. Measure-ments are usually performed at normal incidence, sothat only the in-plane susceptibility is probed. Totake into account the additional layer, we should sim-ply multiply T ab in (10) by a propagation matrix ac-counting for the propagation in the additional layer,and an interface matrix between this layer and thesubstrate. As the matrix T ab is the same in the cur-rent sheet and thin film approaches, the final resultdo not depend on the chosen model, especially at nor-mal incidence for which the isotropic and anisotropicthin film models are equivalent.The optical contrast then depends on the real andimaginary parts of the in-plane susceptibility of the2D material. However, this measurement is stronglydependent on the parameters of the top layer, includ-ing their thickness and permittivity. This explainsprobably the differences in fitting experimental datathat were reported in [18]. We have explored analytically and numerically thelink between the description of a 2D material witha current sheet or thin film model. We have fo-cused our analysis on the description of the intrinsicanisotropy of the layers, i.e. the effect of the out-of-plane component of the single sheet susceptibility orthe out-of-plane component of the dielectric tensor of the thin film. The analytical equivalence in the smallphase shift condition shows that most discrepanciesbetween these two approaches do not come from thefinite thickness of the thin film, but from an incor-rect description of the anisotropy, mainly for TM po-larization and oblique incidence. In particular, wehave shown that considering an isotropic dielectricfunction of a thin film is not equivalent to assumean isotropic single sheet susceptibility. We have alsocommented the fact that a single sheet with vanish-ing out-of-plane response (as graphene) correspondsto an isotropic or an anisotropic effective thin filmdepending of the effective thickness of the film.The application of the transfer matrix approachto classical measurement schemes provides evidencesthat a combination between different techniques isneeded to fully characterize a 2D material, as• transmittance measurements on dielectric sub-strate provide
Im[ χ sx ] at normal incidence andcould provide Im[ χ sz ] at other incidence angle;• standard ellipsometry, in transmission or reflec-tion, cannot separate the in-plane and out-of-plane contributions. However, combined withtransmittance changes it could provide Im[ χ sz ] .Another way to retrieve χ sx and χ sz separatelywould be to perform ellipsometry experimentson different substrates;• optical contrast on a multilayer substrate com-bined with the previous methods can also pro-vide information about Re[ χ sx ] , or even Re[ χ sz ] at oblique incidence.We hope that the present single sheet transfer ma-trix approach, and the analytical connection with thethin film model will help to efficiently perform theanalysis of stratified media involving 2D materials,either for the characterization of 2D materials, or fortheir use in various applications. Acknowledgments
References [1] S. Yu, X. Wu et al. , Advanced Materials ,1606128.[2] F. Bonaccorso, Z. Sun et al. , Nature Photonics , 611 (2010).[3] A. C. Ferrari, F. Bonaccorso et al. , Nanoscale ,4598 (2015).[4] K. F. Mak and J. Shan, Nature Photonics ,216 (2016).[5] M. Bayle, N. Reckinger et al. , Journal of RamanSpectroscopy , 36.[6] D. S. L. Abergel, A. Russell, and V. I. Fal’ko,Applied Physics Letters , 063125 (2007).[7] S. M. Eichfeld, C. M. Eichfeld et al. , APL Ma-terials , 092508 (2014).[8] H. Li, J. Wu et al. , ACS Nano , 10344 (2013).PMID: 24131442.[9] L. Ottaviano, S. Palleschi et al. , 2D Materials ,045013 (2017).[10] T. Stauber, N. M. R. Peres, and A. K. Geim,Phys. Rev. B , 085432 (2008).[11] M. Lobet, B. Majerus et al. , Phys. Rev. B ,235424 (2016).[12] L. A. Falkovsky, Journal of Physics: ConferenceSeries , 012004 (2008). [13] B. Majérus, M. Cormann et al. , 2D Materials ,025007 (2018).[14] F. J. Nelson, V. K. Kamineni et al. , AppliedPhysics Letters , 253110 (2010).[15] S. Cheon, K. D. Kihm et al. , Scientific Reports , 6364 (2014).[16] J. P. Lu, Phys. Rev. Lett. , 1297 (1997).[17] P. Wagner, V. V. Ivanovskaya et al. , Journal ofPhysics: Condensed Matter , 155302 (2013).[18] M. Merano, Phys. Rev. A , 013832 (2016).[19] L. Matthes, O. Pulci, and F. Bechstedt, Phys.Rev. B , 205408 (2016).[20] G. Jayaswal, Z. Dai et al. , Opt. Lett. , 703(2018).[21] I. Valuev, S. Belousov et al. , Applied Physics A , 60 (2016).[22] Y. Li and T. F. Heinz, 2D Materials , 025021(2018).[23] S. Funke, B. Miller et al. , Journal of Physics:Condensed Matter , 385301 (2016).[24] V. G. Kravets, A. N. Grigorenko et al. , Phys.Rev. B , 155413 (2010).[25] A. G. Marinopoulos, L. Reining et al. , Phys.Rev. B , 245419 (2004).[26] E. Dremetsika and P. Kockaert, Phys. Rev. B , 235422 (2017).[27] K. S. Novoselov, A. Mishchenko et al. , Science (2016).[28] F. A. Rasmussen and K. S. Thygesen, The Jour-nal of Physical Chemistry C , 13169 (2015).[29] R. M. A. Azzam, Ellipsometry and polarizedlight (North-Holland Sole distributors for theUSA and Canada, Elsevier Science Pub. Co,Amsterdam New York, 1987).1030] J. E. Sipe, J. Opt. Soc. Am. B, JOSAB , 481(1987).[31] B. U. Felderhof and G. Marowsky, Appl. Phys.B , 11 (1987).[32] R. A. Depine, Graphene Optics: Electromag-netic Solution of Canonical Problems , 2053-2571(Morgan & Claypool Publishers, 2016).[33] M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info . Accessed on2018-07-05.[34] P. Blake, E. W. Hill et al. , Applied Physics Let-ters91